This week in Pre Calc 11 we learned how Solve Rational Equations. We learned that there are multiple ways to solve rational equations, that vary depending on what the arrangement and difficulty of the equation. The first step is to always verify that our terms are completely factored and if they are not, you must factor it and then determine the non permissible values. The first way to solve a rational equation is to move like terms to one side of the equation for example in the equation: you would first want to move the to the right side of the equation to be able to solve it in a more efficient way, because they have a common denominator you are allowed to subract 3 by 2. Another way to solve rational equations is by cross multiplying. However, this method only works if there are only two fractions, one on either side of the equal sign. For exmaple, is an example of an equation that you are able to cross multiply to help you solve it. To cross multiply, you multiply and leaving you with as your new equation. We also learned how to multiply through an equation using a common denominator. When you do this, you are putting the entire equation over a common denominator which means that it can basically cancel out when you multiply through, which will leave you with only the numerator to solve. For example, can be easily solved if you multiply each fraction by the common denominator which is This will leave you with Once we have cancelled out the numerator, you can now solve for The last method that we learned was that if the numerator’s or the denominator’s of an equation that has only two fractions (one on either side of the equal sign) are equal to one another than that means that the numerator’s or denominator’s must also be equal. This allows you to eliminate the numerator’s or the denominator’s to make it easier to solve. For example, have the same denomintor which tells you that the numerator’s will also be equal to one another, allowing you to eliminate the denominator’s, leaving you with a simplified equation of From there you must determine whether or not it’s quadraitc or linear and then you can solve for If it’s quadratic you must make the equation equal to zero and then you must factor the trinomial and find out the possible solutions for and if it’s linear you must isolate for to determine the solution. Once you have determined the solution(s), you must check to make sure that they are not any of the pre-determined non permissible values.

This week in Pre Calc 11 we learned how to add, subtract, divide and multiply rational expressions. We also learned how to determine non-permissible values of rational expressions. In class this week, we reviewed that a rational expression is an algebraic expression that can be written as the quotient of two polynomials. Rational expressions can not have a denominator of zero, therefore the variable in the denominator can not make the denominator equal to zero. The values for the variable that make the variable equal to zero are called the non-permissible values. We may have to factor out the denominator, to be able to determine the non-permissible values.

To multiply rational expressions, we can either multiply straight across, meaning that we multiply the numerators together and the denominators together and then simplify or we may simplify first so that the numbers and variables that we are working with are smaller. For example, in the expression I would first simplify the expression, by eliminating the from the numerator and denominator of this expression and I would also simplify the and to become and Leaving me with the newly simplified expression, Next we are able  to multiply straight across giving us our final answer: This is our final answer because it can not be simplified any further.

To divide rational expressions, we must first flip the numerator and the denominator of the second fraction, then we are able to simplify and then multiply straight across. When writing the non-permissible values of a rational expression that requires division, we must indicate all of the values that the variables can not be, meaning the ones that were in the denominator before it was reciprocated, as well as after.

To add and subtract rational expressions we must first determine the lowest common denominator and then rewrite the fractions as one big fraction, using the lowest common denominator. We may only add or subtract the numerators, the common denominator remains the same. Once we have determined our final answer and reduced it, we must state the non-permissible values.

I have included an example below that shows the detailed steps I would take when adding rational expressions and how to state the non-permissible values.

This week in Pre Calc 11 we learned how to graph Reciprocal Value Functions. A function is reciprocated when the values are flipped, for example the reciprocal of is because the values flipped. In order to graph reciprocal linear functions we must first graph the parent function. Next we will be able to find the horizontal and the vertical asymptotes. In grade 11, the horizontal asymptote will always be drawn along the x-axis, which means the equation for our horizontal asymptote is To determine where the vertical asymptote will be, we must find where our parent function intersects the x-axis, which will be our x-intercept and draw a vertical line through it. This line will be our vertical asymptote, represented by and it’s equation is Once we have determined our asymptotes, we now must find the invariant points. The invariant points are are determined by where the parents function crosses the numbers and on the y-axis. We are now able to graph our reciprocal linear functions by starting at the invariant points and drawing two hyperbola’s that follow along the horizontal and vertical asymptotes, gradually getting closer to them, but never actually touching them.

An example of how to graph a reciprocal linear function is

The first step is to graph the parent function which is is has a slope of and it’s y-intercept is

Next we must find the verical and the horizontal asymptotes. Which will be and because as we can see in the graph, the x-intercept of the parent function is Next we must find the invariant points which will be and Finally, we are able to draw our two hyperbola’s.